Problem: $\int (2x+1)(2x-1)\,dx=$ $+C$
Explanation: The integrand is the product of two functions: $2x+1$ and $2x-1$. Although it is tempting to take the product of their integrals, this would not work. $\int f(x)\cdot g(x)\,dx\neq\int f(x)\,dx \cdot \int g(x)\,dx$ Instead, what we should do is expand the parentheses so we get a nice polynomial. $\int (2x+1)(2x-1)\,dx=\int (4x^2-1)\,dx$ Now we can integrate using the reverse power rule, the sum rule, and the constant multiple rule for indefinite integrals. $\begin{aligned} &\phantom{=}\int (2x+1)(2x-1)\,dx \\\\ &=\int (4x^2-1)\,dx \\\\ &= 4\int x^{2}\,dx -\int 1\,dx \\\\ &=4\dfrac{x^3}{3} -\dfrac{x^1}{1} \dfrac+C \\\\ &=\dfrac{4}{3} x^3 -x +C \end{aligned}$ In conclusion, $\int (2x+1)(2x-1)\,dx=\dfrac{4}{3} x^3 -x +C$